The Copernican Revolution from the Inside

The Copernican revolution was a pivotal event in the history of science. Yet I believe that the lessons most often taught from from this period are largely historically inaccurate and that the most important lessons are basically not taught at all [1]. As it turns out, the history of the Copernican revolution carries important and surprising lessons about rationality — about what it is and is not like to figure out how the world actually works. Also, it’s relevant to deep learning, but it’ll take me about 5000 words on renaissance astronomy to make that point.

I used to view the Copernican revolution as an epic triumph of reason over superstition, of open science over closed dogma. Basically, things went as follows: Copernicus figured out that the sun rather than the earth is at the center of our planetary system. This theory immediately made sense of the available data, undermining its contorted predecessors with dazzling elegance. Yet its adoption was delayed by the Catholic Church fighting tooth and claw to keep the truth at bay. Eventually, with the emergence of Newton’s work and the dawn of the Enlightenment, heliocentrism became undeniable and its adoption inevitable [2].

This view is inaccurate. Copernicus system was not immediately superior. It was rejected by many people who were not puppets of the Church. And among those who did accept it, better fit to the data was not a main reason. What did in fact happen will become clear in a moment. But in reading that, I’d like to prompt you to consider the events from a very particular vantage point: namely what they would be like from the inside. Ask yourself not what these events seem like for a millennial with the overpowered benefit of historical hindsight, but for a Prussian astronomer, an English nobleman or a Dominican priest.

More precisely, there are two key questions here.

First, if you lived in the time of the Copernican revolution, would you have accepted heliocentrism? I don’t mean this as a social question, regarding whether you would have had the courage and resources to stand up to the immensely powerful Catholic Church. Rather, this is an epistemic question: based on the evidence and arguments available to you, would you have accepted heliocentrism? For most of us, I think the answer is unfortunately, emphatically, and surprisingly, no. The more I’ve read about the Copernican revolution, the less I’ve viewed it as a key insight followed by a social struggle. Instead I now view it as a complete mess: of inconsistent data, idiosyncratic mysticism, correct arguments, equally convincing arguments that were wrong, and various social and religious struggles thrown in as well. It seems to me an incredibly valuable exercise to try and feel this mess from the inside, in order to gain a sense what intellectual progress, historically, has actually been like. Hence a key reason for writing this post is not to provide any clear answers — although I will make some tentative suggestions — but to provoke a legitimate sense of confusion.

If things were that chaotic, then this raises the second question. How should you develop intellectually, in order to become the kind of person who would have accepted heliocentrism during the Copernican revolution? Which intellectual habits, if any, unite heliocentric thinkers like Copernicus, Kepler, Galileo and Descartes, and separates them from thinkers like Ptolemy and Tycho? Once again, my answer will be tentative and limited. But my questions, on the other hand, are arguably the right ones.

What happened

My view of the Copernican revolution used to be that when people finally switched to the heliocentric model, something clicked. The data was suddenly predictable and understandable. Something like how Andrew Wiles describes his experience of doing mathematics:

“[…] in terms of entering a dark mansion. You go into the first room and it’s dark, completely dark. You stumble around, bumping into the furniture. Gradually, you learn where each piece of furniture is. And finally, after six months or so, you find the light switch and turn it on. Suddenly, it’s all illuminated and you can see exactly where you were.”

However, this is most certainly not how things appeared at the time. Let’s start at the beginning.

1. Scholasticism

The dominant medieval theory of physics, and by extension astronomy, was Scholasticism, a combination of Aristotelian physics and Christian theology. Scholasticism was a geocentric view. It placed the earth firmly at the center of the universe, and surrounded it with a series of concentric, rotating “crystalline spheres”, to which the celestial bodies were attached.

2. Ptolemy

Ptolemy of Alexandria provided the mathematical foundation for geocentrism, around 100 AD. He wanted to explain two problematic observations. First, the planets appear to move at different speeds at different times, contrary to the Aristotelian thesis that they should move with a constant motion. Second, some planets, like Mars, occasionally seem to briefly move backwards in their paths before returning to their regular orbit. Like this:

In order to explain these phenomena, Ptolemy introduced the geometric tools of equants and epicycles. He placed the earth slightly off the center of the planetary orbits, had the planets themselves orbit in little mini-cycles — so-called “epicycles” — along their original orbit, and introduced another off-center point, called the equant, in relation to which the motions of the planets are uniform, and which Ptolemy also claimed “controlled” the speed of the planets along their larger orbits. Like this:

[3]

Here’s how these additions make sense of retrograde motion [4]:

The ability of the Ptolemaic system to account for these phenomena, predicting planetary positions to within a few degrees (Brown, 2016), was a key contributor to its widespread popularity. In fact, the Ptolemaic model is so good that it’s still being used to generate celestial motions in planetariums (Wilson, 2000).

3. Copernicus

Copernicus published his heliocentric theory while on his deathbed, in 1543. It retained the circular orbits. More importantly, it of course placed the sun at the centre of the universe and proposed that the earth rotates around its own axis. Copernicus was keen to get rid of Ptolemy’s equants, which he abhorred, and instead introduced the notion of an epicyclet (which, to be fair, is kind of just like an equant with its own mini-orbit) [5]. Ptolemy’s system had required huge epicycles, and Copernicus was able to substantially reduce their size.

Retrograde motion falls out of his theory like this:

In order to get the actual motion of the planets correct, both Ptolemy and Copernicus had to bolster their models with many more epicycles, and epicycles upon epicycles, than shown in the above figure and video. Copernicus even considered introducing an epicyclepicyclet — “an epicyclet whose center was carried round by an epicycle, whose center in turn revolved on the circumference of a deferent concentric with the sun as the center of the universe”… (Complete Dictionary of Scientific Biography, 2008).

Pondering his creation, Copernicus concluded an early manuscript outline his theory thus “Mercury runs on seven circles in all, Venus on five, the earth on three with the moon around it on four, and finally Mars, Jupiter, and Saturn on five each. Thus 34 circles are enough to explain the whole structure of the universe and the entire ballet of the planets” (MacLachlan & Gingerich, 2005).

These inventions might appear like remarkably awkward — if not ingenious — ways of making a flawed system fit the observational data. There is however quite an elegant reason why they worked so well: they form a primitive version of Fourier analysis, a modern technique for function approximation. Thus, in the constantly expanding machinery of epicycles and epicyclets, Ptolemy and Copernicus had gotten their hands on a powerful computational tool, which would in fact have allowed them to approximate orbits of a very large number of shapes, including squares and triangles (Hanson, 1960)!

Despite these geometric acrocrabitcs, Copernicus theory did not fit the available data better than Ptolemy’s. In the second half of the 16th century, renowned imperial astronomer Tycho Brahe produced the most rigorous astronomical observations to date — and found that they even fit Copernicus’ data worse than Ptolemy’s in some places (Gingerich, 1973, 1975).

This point seems to have been recognized clearly by enlightenment scholars, many of whom instead chose to praise the increased simplicity and coherence of the Copernican system. However, as just described, it is unclear whether it even offered any such improvements. As Kuhn put it, Copernicus’s changes seem “great, yet strangely small”, when considering the complexity of the final system (Kuhn, 1957). The mathematician and historian Otto Neugebauer writes:

“Modern historians, making ample use of the advantage of hindsight, stress the revolutionary significance of the heliocentric system and the simplifications it had introduced. In fact, the actual computation of planetary positions follows exactly the ancient pattern and the results are the same. […] Had it not been for Tycho Brahe and Kepler, the Copernican system would have contributed to the perpetuation of the Ptolemaic system in a slightly more complicated form but more pleasing to philosophical minds.” (Neugebauer, 1968)

4. Kepler and Galileo

At the turn of the 17th century Kepler, armed with Tycho Brahe’s unprecedentedly rigorous data, revised Copernicus’ theory and introduced elliptical orbits [6]. He also stopped insisting that the planets follow uniform motions, allowing him to discard the cumbersome epicyclical machinery.

Around the same time Galileo invented the telescope. Upon examining the celestial bodies, he found irregularities that seemed to contradict the Scholastic view of the heavens as a perfect, unchanging realm. There were spots on the sun…

…craters and mountains on the moon…

…and four new moons orbiting Jupiter.

Spurred on by his observations, Galileo would soon begin his ardent defense of heliocentrism. Despite the innovations of Galileo and Kepler, the path ahead wasn’t straightforward.

Galileo focused his arguments on Copernicus’ system, not Kepler’s. And in doing so he faced not only the problems with fitting positional planet data, which Kepler had solved, but also theoretical objections, to which Kepler was still vulnerable.

Consider the tower argument. This is a simple thought experiment: if you drop an object from a tower, it lands right below where you dropped it. But if the earth were moving, shouldn’t it instead land some distance away from where you dropped it?

You might feel shocked upon reading the argument, in the same way you might feel shocked by your grandpa making bigoted remarks at the Christmas table, or by a friend trying to recruit you to a pyramid scheme. Just writing it, I feel like I’m penning some kind of crackpot, flat-earth polemic. But if the reason is “well obviously it doesn’t fall like that… something something Newton…” then remind yourself of the fact that Isaac Newton had not yet been born. The dominant physical and cosmological theory of the day was still Aristotle’s. If your answer to the tower argument in any way has to invoke Newton, then you likely wouldn’t have been able to answer it in 1632.

Did you manage to find some other way of accounting for objects falling down in a straight line from the tower? You might want to take a few minutes to think about it.

[…time for thinking…]

Now if at the end of thinking you convinced yourself of yadda yadda straight line physics yadda yadda you were unfortunately mistaken. The tower argument is correct. Objects do drift when falling, due to the earth’s rotation — but at a rate which is imperceptible for most plausible tower heights. This is known as the “Coriolis effect”, and wasn’t properly understood mathematically until the 19th century.

In addition a fair number of astronomical observations seemed to qualitatively contradict heliocentrism — by leaving out predicted phenomena — as opposed to just providing quantitative discrepancies in planetary positions. Consider the stellar parallax. A “parallax” is the effect you might have noticed while looking out of a car window, and seeing how things that are closer to you seem to fly by at a faster pace than things farther away. Like this:

If the earth orbits the sun, something similar should be visible on the night sky, with nearby stars changing their position substantially in relation to more distant stars. Like this:

No one successfully detected a stellar parallax during the renaissance. This included Tycho, who as mentioned above had gathered the most accurate and exhaustive observations to date. His conclusion was that either the distant stars were so distant that a parallax wasn’t detectable using his instruments — which would entail that space was mostly an unfathomably vast void — or there simply was no stellar parallax to be detected.

Once again, with the benefit of hindsight it is easy to arbitrate this debate. Space just is really, really, really vast. But it is worth noticing here the similarity to Russell’s teapot-style arguments [7]. On two points in a row, the defenders of heliocentrism have been pushed into unfalsifiable territory:

Heliocentrist: “There is drift when objects fall from towers — we just can’t measure it!”

Geocentrist: “But provide a phenomenon we can measure, then.”

Heliocentrist: “Well, according to my recent calculations, a stellar parallax should be observable under these conditions…”

Geocentrist: “But Tycho’s data — the best data astronomical we’ve ever had — fails to find any semblance of a parallax. Even Tycho himself thinks the idea is crazy”

Heliocentrist: “The fact that Tycho couldn’t detect it doesn’t mean it’s not there! The stars could be too far away for it to be detected. And things aren’t absurd just because prominent scientists say they’re absurd”

Geocentrist: “Hold on… not only does your new theory contradict all of established physics, but whenever you’re asked for a way to verify it you propose a phenomenon that’s barely testable… and when the tests come out negative you blame the tests and not the theory!”

Heliocentrist: “Okay okay, I’ll give you something else… heliocentrism predicts that Venus will sometimes be on the same side of the sun as Earth, and sometimes on the opposite side…”

Geocentrist: “Yes?”

Heliocentrist: “This means that Venus should appear to vary in size… by…” and the heliocentrist scribbles in his notebook “… as much as… six times.”

And this prediction of the change in size of venus was indeed made by proponents of heliocentrism.

And, once again, although today we know this phenomena does in fact appear, the available observations of the 17th century failed to detect it.

This might all seem messy, complicated, disappointing. If this is what the history of intellectual progress actually looks like, how can we ever hope to make deliberate progress in the direction of truth?

It might be helpful to examine a few thinkers — Copernicus, Kepler, Descartes, Galileo — who actually accepted heliocentrism, and try to better understand their reasons for doing so.

Little is known about the intellectual development and motivations of Copernicus, as the biography written about him by his sole pupil has been lost. Nonetheless, a tentative suggestion is that he developed rigorous technical knowledge across many fields and found himself in environments which were, if not iconoclastic, at least exceptionally open-minded. According to historian Paul Knoll:

“[The arts faculty at the University of Cracow, where Copernicus studied] held the threefold promise of mathematics and astronomy which were abreast of any developments elsewhere in Europe, of philosophical questioning which undermined much the foundations of much that had been characteristically medieval, and of a critical humanistic attitude which was transforming older cultural and educational values” (Knoll, 1975)

Later, when studying law at the University of Bologna, Copernicus stayed with the astronomy professor Domenico Maria Novara, described as “a mind that dared to challenge the authority of [Ptolemy], the most eminent ancient writer in his chosen fields of study” (Sheila, 2015). Copernicus was also a polymath, who studied law in addition to mathematics and astronomy, and developed an early theory of inflation. His pupil Rheticus was an excellent mathematician, and provided crucial support in helping Copernicus complete his final, major work.

Beyond that some authors claim that Copernicus was influenced by a kind of neoplatonism that regarded the sun as a semi-divine entity, being the source of life and energy — which made him more content to place it at the centre of the universe (Kuhn, 1957). These claims are however disputed (Sheila, 2015).

As for Kepler and Descartes, their acceptance of heliocentrism was not motivated by careful consideration of the available data, but commitments to larger philosophical projects. Kepler is known as a mathematician and astronomer, but in his own day he insisted that he be regarded as a philosopher, concerned with understanding the ultimate nature of the cosmos (Di Liscia, 2017). He did have access to better data — Tycho’s observations — than most people before him, and he pored over it with tremendous care. Nonetheless, his preference for elliptical over circular orbits was equally influenced by mystical views regarding the basic geometric harmony of the universe, in which the sun provided the primary source of motive force (Ladyman, 2001; Di Liscia, 2017; Westman, 2001).

Something similar was true of Descartes, although his underlying philosophical agenda is quite different. A striking example of these commitments is that both Kepler and Descartes argued that a heliocentric world-view was self-evident, in the sense of being derivable from first principles without recourse to empirical observation (Frankfurt, 1999).

Beyond that, I know too little about their respective views to be able to offer any more detailed, mechanistic account of why they preferred heliocentrism.

Galileo — Copernicus’ bulldog — is a confusing figure as well. Just like Copernicus, Kepler and Descartes, Galileo was not purely guided by careful experiment and analysis of the data — despite the weight popular history often places upon these characteristics of his. As Einstein writes in his foreword to a modern edition of Galileo’s Dialogue:

“It has often been maintained that Galileo became the father of modern science by replacing the speculative, deductive method with the empirical, experimental method. I believe, however, that this interpretation would not stand close scrutiny. There is no empirical method without speculative concepts and systems; and there is no speculative thinking whose concepts do not reveal, on closer investigation, the empirical material from which they stem.” (Einstein, 2001)

For Galileo, this speculative system consisted in replacing the four Aristotelian elements with a single, unified theory of matter, and replacing the view of nature as a teleological process with a view of it a deterministic, mechanistically intelligible process. Einstein later points out that in some respects this approach was inevitable given the limited experimental methods available to Galileo (for example, he could only measure time intervals longer than a second).

Galileo was also a man of courage and belligerence. One of his strengths was an absolute refusal to accept arguments from authority without experimental evidence or careful reasoning. It appears as if though his belligerence aided him several times in a quite ironic way. Many of the arguments he marshalled against his opponents were either incorrect, or correct but based on incorrect observations. One example is his attempt to derive a theory of tides from the motions of the earth, a project to which he devotes about a fourth of his famous Dialogue. Einstein, again, writes “it was Galileo’s longing for a mechanical proof of the motion of the earth which misled him into formulating a wrong theory of the tides. [These] fascinating arguments […] would hardly have been accepted as proofs by Galileo, had his temperament not got the better of him”.

Moreover, Galileo’s observations of sunspots and moon craters weren’t unproblematic. In both cases there is evidence to indicate that he was fooled by optical illusions. And though he was also right about the existence of moons orbiting Jupiter, which contradicted the uniqueness of the earth as the only planet with a moon, what he actually observed rather seems to have been Saturn’s rings (Ladyman, 2001) [8].

Nonetheless, at this point you might be aching to object that, disregarding inconsistent data, theoretical flaws, failed predictions and incorrect formulation of a theory of the tides… surely Galileo’s Dialogue provided other convincing arguments that finally tipped the balance in favour of heliocentrism?

Alas, history is messy.

Recall that Galileo defended Copernicus system, not Kepler’s, and hence had to deal with its flaws. More strikingly, in the above I still haven’t mentioned the existence of a third major theory, rivalling both Ptolemy and Copernicus: Tycho Brahe’s combined geoheliocentric theory. This theory retained the moon and sun in orbit around the earth but placed all the other planets in orbit around the sun.

Galileo’s Dialogue does not engage with Tycho’s theory at all. One suggested explanation (given by an unknown Wikipedia contributor) is that, assuming Galileo’s theory of the tides, the Ptolemaic and Tychonic systems are identical, and hence it would suffice to rebut the former. But the theory of the tides was wrong.

These theories only differ in their prediction of whether we should be able to observe stellar parallaxes. And as mentioned above, Tycho’s data had failed to detect one, which he saw as key evidence for his view.

Eventually though, this historical mess was straightened out, and a crucial experiment arbitrated Galileo, Tycho and Ptolemy. German astronomer Friedrich Bessel’s finally managed to observe a stellar parallax in 1838. About 200 hundred years later. By that point, the Copernican revolution was surely already over — even the Catholic church had removed Copernicus’ De revolutionibus from Its index of banned books, as it was simply accepted as true (Lakatos & Zahar, 1975).

5. Newton

At one point Newton also came along, but Galileo died about a year before he was born. Newton’s marriage of physics and mathematics, which implied Kepler’s laws as a special case, was crucial in demonstrating the viability of heliocentrism. But nonetheless some thinkers did something very right decades before the arrival of the Cambridge genius, which he was very well aware of. For the Copernican revolution might have been completed by Newton, but in the end he still stood on the shoulders of giants.

Now what?

One purpose of this essay has been to portray an important historical era in a more realistic way than other popular portrayals. I asked two questions at the beginning:

If you lived in the time of the Copernican revolution, would you have accepted heliocentrism?

How should you develop intellectually, in order to become the kind of person who would have accepted heliocentrism during the Copernican revolution?

The preceding section argued that the answer to the first question might quite likely have been no. This section takes a closer look at the second question. I do however want to preface these suggestions by saying that I don’t have a good answer to this myself, and suggest you take some time to think of your own answers to these questions. I’d love to hear your thought in the comments.

What about Ibn ash-Shãtir?

There seems to be some Islamic scholars who beat Copernicus to his own game by a few hundred years. I’d be keen to learn more about their story and intellectual habits.

Careful with appearances

Geocentrists liked to claim that it certainly seems like the sun orbits the earth, and not vice versa. There is something odd about this. Consider the following Wittgenstein anecdote:

“He [Wittgenstein] once asked me [Anscombe]: ‘Why do people say it is more logical to think that the sun turns around the Earth than Earth rotating around its own axis?’ I answered: ‘I think because it seems as if the sun turns around the Earth.’ ‘Good,’ he said, ‘but how would it have been if it had seemed as if the Earth rotates around its own axis then?’” (Anscombe, 1959)

This quote hopefully inspired in you a lovely sense of confusion. If it it didn’t, try reading it again.

When I said above that it certainly seems like the sun orbits the earth and not vice versa, what I meant to say was that it certainly seems like it seems likethe sun orbits the earth and not vice versa [9].

There’s a tendency to use the word “seems” in quite a careless fashion. For example, most people might agree that it seems like, if an astronaut were to push a bowling ball into space, it would eventually slow down and stop, because that’s what objects do. At least most people living prior to the 20th century. However, we, and they, already know that this cannot be true. It suffices to think about the difference between pushing a bowling ball over a carpet, or over a cleaned surface like polished wood, or over ice — there’s a slippery slope here which, if taken to its logical extreme, should make it seem reasonable that a bowling ball wouldn’t stop in space. A prompt I find useful is to try to understand why the behaviour of the bowling ball in space could not have been any other way, given how it behaves on earth. That is, trying to understand why, if we genuinely thought a bowling ball would slow down in space, this would entail that the universe was impossibly different from the way it actually is.

Something similar seems true of the feeling that the sun orbits the earth, and this is brought out in the Wittgenstein anecdote. What we think of as “it seems as if though the sun orbits the earth” is actually just us carelessly imposing a mechanism upon a completely different sensation, namely the sensation of “celestial objects seeming to move exactly as they would move if the earth orbited the sun and not vice versa”. Whatever it would look like to live in a world where the opposite was true, it certainly wouldn’t look like this.

Careful with your reductios

Many of the major mistakes made by opponents of heliocentrism was to use reductio ad absurdum arguments without really considering whether the conclusion was absurd enough to actually overturn the original argument. Tycho correctly noted that either there wasn’t a stellar parallax or he couldn’t measure it, but incorrectly took the former as more plausible. Proponents of the tower argument assumed that objects fall down in straight lines without drift, and that anything else would be percetible by the naked eye. In both cases, people would just have been better off biting the bullet and accepting the implications of heliocentrism. That, of course, raises the question of which bullets one should bite — and that question is beyond the scope of this essay.

The data is not enough

There’s a naïve view of science according to which the scientist first observes all the available data, then formulates a hypothesis that fits it, and finally tries to falsify this new hypothesis by making a new experiment. The Copernican revolution teaches us that the relation between data and theory is in fact much more subtle than this.

A true theory does not have to immediately explain all the data better than its predecessors, and can remain inconsistent with parts of the data for a long time.

The relation between data and theory is not a one-way shooting range, but an intricate two-way interplay. The data indicates which of our theories are more or less plausible. But our theories also indicate which data is more or less trustworthy [10]. This might seem like a sacrilegious claim to proponents of the naïve view described above: “ignore the data!? That’s just irrational cherry-picking!” Sure, dishonest cherry-picking is bad. Nonetheless, as the Copernican revolution shows, the act of disregarding some data in a principled manner as it doesn’t conform to strong prior expectations has been critical to the progress of science [11].

When Einstein famously remarked “God doesn’t play dice”, he arguably adopted the same kind of mindset. He had built a complex worldview characterised by a certain mathematical law-likeness, and was confident in it to the extent that if quantum mechanics threatened its core principles, then quantum mechanics was wrong — not him.

Sometimes, scientists have to be bold — or arrogant — enough to trust their priors over the data.

…and, finally, deep learning

It seems apt to notice some similarities between the state of astronomy during the Copernican revolution and the current state of deep learning research.

Both are nascent fields, without a unifying theory that can account for the phenomena from first principles, like Newtonian physics eventually did for astronomy.

Both have seen researchers cling to their models for decades without encouraging data: many of the most succesful current deep learning techniques (conv nets, recurrent nets and LSTMs, gradient descent, …) were invented in the 20th century, but didn’t produce spectacular results until decades later when sufficient computing power became available. It would be interesting to find out if people like like Geoffrey Hinton and Yann Le Cunn share intellectual habits with people like Copernicus and Galileo.

Finally, I’m particularly struck by the superficial similarities between the way Ptolemy and Copernicus happened upon a general, overpowered tool for function approximation (Fourier analysis) that enabled them to misleadingly gerrymander false theories around the data, and the way modern ML has been criticized as an inscrutable heap of linear algebra and super-efficient GPUs. I haven’t explored whether these similarities go any deeper, but one implication seems to be that the power and versatility of deep learning might allow suboptimal architectures to perform deceivingly well (just like the power of epicycle-multiplication kept geocentrism alive) and hence distract us from uncovering the actual architectures underlying cognition and intelligence.

Footnotes

[1] They of course are taught, because that is how I learnt about them. But this was in a university course in the philosophy of science. The story of Galileo is probably taught in most middle schools [no source, my own hunch]. But only about 0.5% of US college students major in philosophy [source], and I’d guesstimate something like a third of them to take classes in philosophy of science.

[2] This last step is kind of a blackbox. My model was something like “a true theory was around for long enough, and gained enough support, that it was eventually adopted”. This sounds quite romantic, if not magical. It’s unclear exactly how this happened, and in particular what strategic mistakes the Church made that allowed it to.

[3] Figure credit of the Polaris Institute of Iowa State University, which provides a great tutorial on medieval and renaissance astronomy here.

[4] As the planets orbit the center, they will also orbit the equant in a circular fashion. The key difference here is the speed of motion. Think about it: the fact that the equant is not located at the center of their orbits means that I spend way too long trying to understand this, but this animation was helpful.

[5] I spent two hours trying to understand the geometry of this and I won’t drag you down that rabbit-hole, but if you’re keen to explore yourself, check out these links: [1], [2].

[6] It is however a common mistake to imagine these as clearly elongated ellipses: their eccentricity is very small. For most practical purposes apart from measurement and prediction they look like circles (Price, 1957).

[7] Russell’s teapot is a skeptic thought-experiment intended to reveal the absurdity of unfalsifiable views, by postulating that there’s a teapot orbiting Jupiter and that it’s too small to be detectable using current instruments, but nonetheless insisting that it really is there.

[9] There’s of course a sense of mysticism in this, which — like the rest of Wittgenstein’s mysticism — I don’t like. Mysticism is mostly just a clever way of scoring social understanding-the-world-points without actually understanding the world. It might be that heliocentrism and geocentrism are genuinely indistinguishable from our vantage point, in which case the confusion here is just a linguistic sleight-of-hand, rather than an actual oddity in how we perceive the world. But this doesn’t seem correct. After all, we were able to figure out heliocentrism from ourvantage point, indicating that heliocentrism is distinguishable from geocentrism from our vantage point.

[10] In Bayesian terms, your posterior is determined by both your likelihoods and your priors.